Sampling and recovery of pulse streams

Chinmay Hegde, Richard G. Baraniuk

Research output: Contribution to journalArticlepeer-review

39 Scopus citations


Compressive sensing (CS) is a new technique for the efficient acquisition of signals, images and other data that have a sparse representation in some basis, frame, or dictionary. By sparse we mean that the N-dimensional basis representation has just K ≪ N significant coefficients; in this case, the CS theory maintains that just M = O(K log N) random linear signal measurements will both preserve all of the signal information and enable robust signal reconstruction in polynomial time. In this paper, we extend the CS theory to pulse stream data, which correspond to S-sparse signals/images that are convolved with an unknown F-sparse pulse shape. Ignoring their convolutional structure, a pulse stream signal is K=SF sparse. Such signals figure prominently in a number of applications, from neuroscience to astronomy. Our specific contributions are threefold. First, we propose a pulse stream signal model and show that it is equivalent to an infinite union of subspaces. Second, we derive a lower bound on the number of measurements M required to preserve the essential information present in pulse streams. The bound is linear in the total number of degrees of freedom S + F, which is significantly smaller than the nave bound based on the total signal sparsity K=SF. Third, we develop an efficient signal recovery algorithm that infers both the shape of the impulse response as well as the locations and amplitudes of the pulses. The algorithm alternatively estimates the pulse locations and the pulse shape in a manner reminiscent of classical deconvolution algorithms. Numerical experiments on synthetic and real data demonstrate the advantages of our approach over standard CS.

Original languageEnglish (US)
Article number5677611
Pages (from-to)1505-1517
Number of pages13
JournalIEEE Transactions on Signal Processing
Issue number4
StatePublished - Apr 2011


  • Blind deconvolution
  • compressive sensing
  • sparsity
  • union of subspaces

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Signal Processing


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